Anisotropic
5.2 Model Formulation
The complex, hierarchical nature of VACNTs makes the choice of scale in modeling their behavior non-trivial. At magnifications of 1000×they appear as arrays of vertically aligned tubes, markedly anisotropic. Magnifying one hundred times more reveals their highly inter- connected, foam-like structure and the network of CNTs begins to appear nearly isotropic.
Magnifying another hundred times one obtains a view of the individual CNTs themselves.
In this work, we propose a model that smooths over the discrete nature of the individ- ual tubes and approximates the overall material behavior through an isotropic continuum constitutive relation in the same spirit as, for example, in the Deshpande-Fleck constitu- tive relation for foams [51]. Our model differs from others for VACNTs that focus on the scale at which the material appears to be a nominally aligned array of tubes [14, 15, 24].
The complex deformation and stress-strain behavior observed in Chapter 4 (Hutchens et al.
[44]) serves as both motivation and validation for the choice of constitutive relation.
We use a finite deformation formulation and express the constitutive relation in terms of the rate of deformation tensor,d, the symmetric part ofF·˙ F−1, whereFis the deformation gradient, and the Kirchhoff stress τ = Jσ, with σ being the Cauchy stress and J = det(F). A superposed dot denotes the partial derivative with respect to time. The rate of deformation tensor is taken to be the sum of elastic, de, and plastic, dp, parts. Elastic strains are assumed to be small and are given by
de = 1 +ν E τˆ− ν
Etr(ˆτ)I, (5.1)
whereE is Young’s modulus,ν is Poisson’s ratio, tr(·)denotes the trace, Iis the identity tensor andτˆ is the Jaumann rate of Kirchhoff stress.
In the experiments in Chapter 4 (Hutchens et al. [44], summarized in Fig. 5.1), little recovery of deformation was observed so that a material model framework allowing for permanent deformation was used. Also, material rate dependence is taken into account both for numerical reasons as well as in accordance with the observations by Zhang et al.
[47]. We model the irrecoverable deformation response by a modification of the relation for an isotropic, hardening viscoplastic solid to account for the compressibility of the VACNTs.
We write
dp= 3 2
˙ p
σe[s+Btr(τ)I] (5.2)
with
˙ p = ˙0
σe g
1/m
. (5.3)
Here, ˙0 is a reference strain rate, m is the rate hardening exponent, s is the deviatoric
Kirchhoff stress tensor,s=τ−tr(τ)I/3, andσeis the equivalent stress,σe =p
3/2s:s.
The compressibility parameterBis specified in terms of a plastic Poisson’s ratio,νp, by B = 1
3
1−2νp 1 +νp
. (5.4)
h3
h2
h1
Figure 5.2: A plot of the hardening function, g(p), for 1 = 0.005, 2 = 0.1, h1 = 5.0, h2 =−5.0, andh3 = 1.5illustrating its general shape as defined in Eq.(5.5).
Motivated by structural load-deflection responses that give rise to periodic folds (e.g., [52–54]), we characterize the flow strength or hardening function, g(p), as consisting of a hardening range followed by softening and then subsequent rehardening. A simple form that embodies these features is
g(p) σ0 =
1 +h1p p < 1
1 +h11+h2(p−1) 1 < p < 2 1 +h11+h2(2−1) +h3(p−2) p > 2
, (5.5)
and is depicted in Fig. 5.2. Parameters h1, h2, and h3 determine the hardening and soft- ening slopes and1 and2 are the strains at which the hardening-softening and softening- hardening transitions occur, respectively, andσ0is a reference stress. The simplified, piece- wise nature of the flow strength curve lends itself well to a systematic study of changes in behavior with variations in its shape as discussed in Section 5.3. It is worth noting that
the presence of a material rate dependence acts to regularize the governing equations in the softening regime [55]. A similarly shaped stress-strain relation, that of a bi-stable spring, was used by Fraternali et al. [50] as a microscale element of a 1-D model that captured the quantitative aspects of a reversible deformation response in uniaxially loaded bulk VACNT samples [24]. Attention is restricted to axisymmetric deformations, which eliminates the ability of the model to capture the nucleation and lateral buckle propagation seen in exper- iments (Fig. 5.1(b)), but still allows for local sequential buckling and significantly reduces the computational time.
The finite element formulation is based on the dynamic principle of virtual work, which can be written as
Z
V
τ :δddV = Z
S
T·δu˙ dS− Z
V
ρ¨uδudV, (5.6)
whereV andSare, respectively, the volume and surface of the body in the initial configu- ration,Tis the traction vector, anduis the displacement vector.
We perform calculations for a cylinder of heightHand radiusR. With the assumption of axisymmetric conditions in a cylindrical coordinate system (r, θ, z) all field quantities are independent ofθ. A velocityu˙z(t)is imposed at the top of the pillar,z=H, with
˙
uz(r, H, t) =
−vztt
rise fort < trise
−vz fort > trise
, (5.7)
andTr(r, H) = 0. Here,trise is the time interval over which the velocity is ramped up to avoid shock loading the system. The bottom of the pillar is presumed fixed to the substrate sou˙r(r,0, t) = ˙uz(r,0, t) = 0. The outer surface of the pillar is taken to be traction free, Tr(R, z) = 0. We do not account for possible contact between the folds that develop due to buckling. The calculations are terminated prior to any material contact.
The finite element discretization of Eq. (5.6) is based on a convected coordinate rep- resentation of the governing equations with linear displacement crossed triangles as in a number of previous analyses, e.g., Tvergaard et al. [56] and Tvergaard and Needleman [57]. Time integration is carried out by the explicit Newmarkβ-method [58] using lumped masses. The rate tangent method of [59] is used for the constitutive update.
5.2.1 Simulation Parameters
The calculations are carried out withE/σ0 = 100,m = 0.02(on the order of experimen- tally measured values of rate sensitivity in VACNTs [47]),˙0 = ˙ref,ν = 0.25, νp = 0.25.
The mesh geometry is that of a circular cylindrical pillar with an aspect ratio, H/R, of3.
The imposed velocity,vz, in Eq. (5.7) is fixed atκ˙refHwithκ = 0.004and the ramp time trise = 5/˙ref. The initial hardening portion of Eq. (5.5) was fixed at1 = 0.005, h1 = 5 throughout this study. Results are presented for variations in2,h2,h3.
The finite element mesh in all calculations consists of a uniform 80× 240 mesh of quadrilateral “crossed triangle” elements each of which isH/240×H/240.
If the analyses were quasi-static, these dimensionless parameters would be sufficient to characterize the formulation. However, dynamic, rather than quasi-static, analyses are carried out because, even though the response is generally quasi-static, dynamic snapping can occur due to the up-down-up shape ofg(p)(see Fig. 5.2). Hence, a density needs to be specified and is taken to beρ= 10−14σ0( ˙ref/H)2in non-dimensional form.
For σ0 = 0.1 MPa, H = 75 µm and ˙ref = 25 s−1, we have E = 10 MPa and ρ = 1.11×10−4 MPa/(m s)2 (ρ = 111 kg/m3). Also, vz = 7.5 µm/s and trise = 0.2, which corresponds to the applied displacement achieving its constant value when the over- all strain,uz(r, H, t)/H, reaches0.01. All of these values are of a similar order to those in the experiments.
Axial gradients in E and σ0 are incorporated into the material through multiplication of these variables by a dimensionless functionQ(z), whereQ(z) ≡ 1corresponds to the case in which there is no gradient withz evaluated at the center of the element for which the rescaledEandσ0values are being calculated.